We continue with the proof of Theorem REF  which relaxes the requirement of Theorem REF  that all the arcs of \([X_1,Y]\)  have distinct labels, all the arcs of \([Y,X_2]\)  have distinct labels and all the arcs of \([X_1,X_2]\)  have distinct labels.
In Figure REF , containing a non-trivial complete bipartite subgraph of \(G\)  for which all arcs have identical labels, we have shown in a simple example how a graph \(G\)  can be decomposed into the graphs \(G/Y\)  and \(G/X_1/X_2\)  such that \(G\cong G/YG/X_1/X_2\) .
In Theorem REF , we use the proof of Theorem REF  given in [1]} and modify this proof to support complete bipartite subgraph of \(G\)  with arcs with the same label of \([X_1,Y]\)  and \([Y,X_2]\)  and bipartite subgraphs of \(G\)  with arcs with the same label of \([X_1,X_2]\) .
Note that a bipartite subgraph of \(G\)  arc-induced by all arcs with the same label of \([X_1,X_2]\)  does not have to be complete.
